Prior to the present invention various bracket constructions have been devised and utilized to mount automotive accessories to the engine so that they can be driven thereby and are accessible for service. Examples of such prior art are found in the following U.S. Pat. No. 3,730,147 to Buckwald for Engine Accessory Arrangement; U.S. Pat. No. 5,065,713 to W. Seats for Mounting Brackets For Mounting Engine Accessories; and U.S. Pat. No. 4,633,828 to L. Steele for Adjustable Polar Point Mount. While these prior art arrangements provide various constructions for accessory mounting and generally meet their objectives, they do not meet new and higher standards for attaching and supporting an accessory at a first point directly to the engine and then providing axial shifting of the accessory and mount for aligning the accessory for final attachment to the engine by at least one fastener. Moreover and in contrast to the present invention, the prior art mounting constructions and methods are not readily adaptable for accommodating relatively large tolerances common in mass produced components. Also, they do not conveniently and rigidly anchor the accessory at multiple attachment points to the engine so that the accessory is permitted to be axially shifted for positioning in a predetermined desired location for subsequently receiving a fastening member which finishes the attachment of the accessory to the engine and which is capable of handling relatively great torque inputs by a drive belt driven by the engine crankshaft. Furthermore, the prior constructions do not provide for material savings of engine compartment space and for self-locking and self-aligning with respect to the supporting structure.